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Zero localization and asymptotic behavior of orthogonal polynomials of Jacobi-Sobolev. (English) Zbl 1046.33008

Summary: We consider the Sobolev orthogonal polynomials associated to the Jacobi’s measure on \([-1,1]\). It is proved that for the class of monic Jacobi-Sobolev orthogonal polynomials, the smallest closed interval that contains its real zeros is \([-\sqrt{1+2C}, \sqrt{1+2C}]\) with \(C\) a constant explicitly determined. The asymptotic distribution of those zeroes is studied and also we analyze the asymptotic comparative behavior between the sequence of monic Jacobi-Sobolev orthogonal polynomials and the sequence of monic Jacobi orthogonal polynomials under certain restrictions.

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
41A10 Approximation by polynomials
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46G10 Vector-valued measures and integration
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